Can i "execute" the Compositor node graph without re-rendering?
If I understand your question, when you render an image and set "use nodes" in the compositor, you can check the "backdrop" function, then Ctrl Shift click on a node to create a viewer and see the output of that node on the backdrop. You can see the result anlso in an UV/image editor, if you change the header's dropdown menu from "render result" to "Viewer node".

As soon as you change a value in the compositor, the change is reflected in the Viewer (Backdrop or UV/Image window), without any need to push any button to trigger the compositor, and without the need of redoing the render: it's automatic (maybe a little slow and not realtime: check the upper info header for a compositing Time bar)